Trigonometic Limit (tan x)^tan(x + Pi/4)

[e_2.718281828]

I really need help here. (use screenshot pls)
To solve a limit, I always first try to put in the value we're looking at( for this exercise, it's x->Pi/4)
If this gives me an undetermined form, I try to use L'Hopital's rule. However, in this exercise, we do not have a 0/0 or oo/oo solution...
Since we have tan(x + Pi/4) I can use a trigonometric formula that tells me that tan(A + B) = (tan A + tan B)/(1-tan A. tan B)
Still undetermined forms so I use logarithms to make the problem perhaps easier.
f(x)^g(x) = e^(ln(f(x))).g(x), and still nothing. I really hope someone can help me here, it's the last exercise of the series, and I got the first 4 pretty quickly....

 

[tkhunny]

I really need help here. (use screenshot pls)
To solve a limit, I always first try to put in the value we're looking at( for this exercise, it's x->Pi/4)

You should not be doing that - unless you first go to the trouble to prove it continuous at that point. It's a limit. You don't get to arrive. You are interested in approaching behavior, not arrival behavior.

If this gives me an undetermined form, I try to use L'Hopital's rule. However, in this exercise, we do not have a 0/0 or oo/oo solution...
Since we have tan(x + Pi/4) I can use a trigonometric formula that tells me that tan(A + B) = (tan A + tan B)/(1-tan A. tan B)
Still undetermined forms so I use logarithms to make the problem perhaps easier.
f(x)^g(x) = e^(ln(f(x))).g(x), and still nothing. I really hope someone can help me here, it's the last exercise of the series, and I got the first 4 pretty quickly....
View attachment 9143

Perhaps you should use a logarithm transformation, rather than whatever that third expression is that I can't understand.

\(\displaystyle \lim_\limits{x\rightarrow a}f(x) = L, \iff\lim_\limits{x\rightarrow a}log(f(x)) = log(L)\) - assuming everything exists.

 

[Dr.Peterson]

I really need help here. (use screenshot pls)
To solve a limit, I always first try to put in the value we're looking at( for this exercise, it's x->Pi/4)
If this gives me an undetermined form, I try to use L'Hopital's rule. However, in this exercise, we do not have a 0/0 or oo/oo solution...
Since we have tan(x + Pi/4) I can use a trigonometric formula that tells me that tan(A + B) = (tan A + tan B)/(1-tan A. tan B)
Still undetermined forms so I use logarithms to make the problem perhaps easier.
f(x)^g(x) = e^(ln(f(x))).g(x), and still nothing. I really hope someone can help me here, it's the last exercise of the series, and I got the first 4 pretty quickly....
View attachment 9143

I think your third line is essentially tkhunny's logarithm transformation. What you've found it that the log (your exponent) has the form 0 * infinity. There are standard ways to deal with this, by transforming it into either 0/0 or infinity/infinity. I don't see that either will be particularly nice, but it's worth trying. For example, if f(x) -> 0 and g(x) -> oo, you can rewrite f(x) * g(x) as f(x) / (1/g(x)) -> 0/0, or as g(x)/(1/f(x)) -> oo/oo.

I don't know whether your use of the angle sum helps or not, in the long run. You might try it both ways.

 

[e_2.718281828]

You should not be doing that - unless you first go to the trouble to prove it continuous at that point. It's a limit. You don't get to arrive. You are interested in approaching behavior, not arrival behavior.



Perhaps you should use a logarithm transformation, rather than whatever that third expression is that I can't understand.

\(\displaystyle \lim_\limits{x\rightarrow a}f(x) = L, \iff\lim_\limits{x\rightarrow a}log(f(x)) = log(L)\) - assuming everything exists.

At school the teacher always says to check with the value, otherwise you can't know if you can apply l'hopital. Also sometimes you just get +oo for example, and it's not worth factoring and deriving for 15 minutes to get an answer you could have gotten straight away

 

[tkhunny]

I think your third line is essentially tkhunny's logarithm transformation.

Have to disagree with this assessment. The OP has modified the expression with the neutral introduction of a logarithm and an exponent. This has resulted in nothing. Whoever suggested this alteration failed also to suggest that it should lead somewhere. In my opinion, the variation I offered suggests much more clearly that something should happen after the transformation.

 

[tkhunny]

At school the teacher always says to check with the value, otherwise you can't know if you can apply l'hopital. Also sometimes you just get +oo for example, and it's not worth factoring and deriving for 15 minutes to get an answer you could have gotten straight away

That makes no sense. What does "check with the value" even mean?

What "15 minutes"? You should have a sufficient background in the manipulation of logarithms. \(\displaystyle \log\left(a^{b}\right) = b\cdot\log(a) = \dfrac{log(a)}{\small{\dfrac{1}{b}}}\). This is not calculus. Don't let the algebra get in the way of your learning of the calculus.

 

[e_2.718281828]

That makes no sense. What does "check with the value" even mean?

What "15 minutes"? You should have a sufficient background in the manipulation of logarithms. \(\displaystyle \log\left(a^{b}\right) = b\cdot\log(a) = \dfrac{log(a)}{\small{\dfrac{1}{b}}}\). This is not calculus. Don't let the algebra get in the way of your learning of the calculus.

What I mean by this is that for example in
lim(x->pi/2) (ln x)/(cos x) = (ln pi/2)/0 (indetermined form)
+oo at 0+
-oo at 0-